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Posted: February 9th, 2023

ECON 100C—Homework 2

Exercise 1: TFP in the production and Solow models. (7 points) To solve this exercise you will need to

use the data contained in the file “Data_HW2”. It contains cross-country information on GDP and

Capital stock in constant dollars. Also, it contains information on Population and the average value

of human capital per person.

Math refresh

• Let X and Z be two variables and a and b two constants. Then the variance of aX + bZ is equal

to

var(aX + bZ) = var(aX) + var(bZ) + 2cov(aX, bZ)

= a

2var(X) + b

2var(Z) + 2ab cov(X, Z),

where cov(X, Z) is the covariance of X and Z.

• Rule of logarithms: log(aXb

) = log(a) + b log(X).

Now we can start the exercise.

a. This first question involves several steps. First, you will need to build a measure of total labor

input L. This will be defined as the total human capital in the economy. So if we have the

average of a human capital index per person and population in the data, just multiply these

two series to obtain our labor input L. Now, construct output per labor input y ≡ Y/L, where

Y represents GDP. Similarly, denote capital er unit of labor input k. Construct also k. Now,

recall out production model. We had that output per labor input was equal to

y = Ak1/3, (1)

where A is the Total Factor Productivity, or TFP. Now using the data on y and k compute A

according to equation 1.

Take the natural logarithm of y and do a variance decomposition according to equation 1.

1

(Take the log of that expression and you should end with log y being a linear function of log A

and log k).

1Pay attention when you do this. You should use the natural logarithm, so beware of the base your software is using. Sometimes the

default in some softwares is logarithm with a base of 10.

1

What percentage of the variance of log y is explained by the term containing log A? What

about the term containing log k? And the covariance term? (1 point)

b. Will the results in the variance decomposition would have changed if we were using billions

of dollars for capital and output instead of millions? What about if would have normalized

the whole L variables such that the L corresponding to the US is equal to 1? Explain your

answer. (0.5 points)

c. Let s be the savings rate. Compute cov(log(s), log(A)). Keep it handy. Now recall the expres sion for capital per labor unit in the steady state of the Solow model

k =

sA

d

3/2

, (2)

where d is the depreciation rate. We are going to use this expression to unpack cov(log A, 1/3 log k).

Assume d is constant across countries. Using 2, what is the condition regarding s and A such

that cov(log A, 1/3 log k) > 0. Is it satisfied empirically? (Hint: take logs of equation 2!). (1.5

points)

d. Substitute the expression for capital per labor unit 2 into the output per labor unit expression

1. Assume d is the same across countries. Now taking logs of y and using the previously

calculated log A, do a variance decomposition that should depend on log A and log s. What

percentage is explained by the term containing log A? What about the term containing log s?

And the covariance term? (1 point)

e. Why is the percentage of variance explained by the variance and covariance terms containing

log A and log s not equal to 100 %? Could we use this difference as some informal test of the

Solow model? Explain. (0.5 points)

f. Use the new expression on y after substituting equation 2 into equation 1 and the data on y

and s to compute a new series on TFP. For this purpose, and consistent with the assumption

of constant depreciation rates, we can normalize that d = 1. Denote this new TFP series as As

.

Compute the correlation between log A and log As

. Is it high or low? What does it mean in

terms of TFP ranking across countries with the two methods? (0.5 points)

g. Do again a variance decomposition as in sub-question d but using log As

instead of log A.

What percentage is explained by the two variance terms and the covariance term? Taking

stock of the different variance decompositions you have made, what would you conclude

about the importance of TFP in explaining GDP per labor unit? (0.5 points)

h. Why the Solow model helped us unpack the importance of TFP that was hidden in cov(log A, log k)?

(1 point)

Exercise 2: The Solow Model and Population growth. (3 points)

a. The original Solow model we saw in class has 5 equations (pero pertiod t) for five variables:

output Y, capital K, labor L, consumption C, and investment I. Take any variable X. Then its

2

corresponding lower case variable x is defined as x ≡ X/L, i.e. X’s per capita corresponding

value.2 We will use a little more general notation than in class, so the production function is

equal to

Y = AKαL

1−α

.

Characterize the equilibrium of the model at any period t for the per capita variables. (Hint:

it should be now only 4 equations). (1 point).

b. Now assume the depreciation rate d = 0. Explain why the model has or does not have a

steady state with k > 0. (0.5 points)

c. We assume now that there is population growth. In particular we assume it grows at a constant

rate γ. This means that

Lt+1 = (1 + γ)Lt

.

Rewrite the equilibrium of the model at time t in per capita terms at any period t. For this case

you can take Kt+1 as a variable and Kt as a parameter (or state variable) at period t.

3

. (Hint:

Don’t use the delta notation in the capital accumulation equation). (0.5 points).

d. Again assume that the depreciation rate is equal to 0. Explain why the model has or does not

have a steady state (for the per capita variables) with k > 0. (1 point).

2For example GDP per capita is y and capital per capita is k.

3This means that Kt should be taken as given, or exogenous, in period t

3

a. To compute Total Factor Productivity (TFP), the production model must be used: y = Ak^(1/3), where y is output per labor input, k is capital per unit of labor input, and A is TFP. To obtain labor input, multiply the average of human capital index per person by population. To calculate TFP, solve for A in the equation by dividing Y (GDP) by L (labor input). Then, take the natural logarithm of y and perform a variance decomposition as outlined in the math refresh section. The percentage of variance of log y explained by log A, log k, and the covariance term can be found by performing the calculation.

b. No, the results would not change if billions of dollars were used instead of millions or if the L variable was normalized such that the L corresponding to the US is equal to 1. The results would change if the scales of the variables were changed, but normalizing or changing units does not affect the relationship between the variables.

c. To find the condition for cov(log A, 1/3 log k) > 0, take the natural logarithm of the steady state expression for capital per labor unit in the Solow model (k = (sA/d)^(3/2)), then compute cov(log A, 1/3 log k). The condition is that s and A have a positive correlation, which can be tested empirically.

d. Substitute the expression for capital per labor unit into the output per labor unit expression, then take the natural logarithm of y. Perform a variance decomposition that depends on log A and log s. The percentage of variance explained by log A, log s, and the covariance term can be found.

e. The percentage of variance explained by log A and log s is not equal to 100% because other factors, such as depreciation rate (d), technology, and efficiency, may also affect output. The difference could be used as an informal test of the Solow model, but it would not be a definitive test as other factors may also be affecting output.

f. To calculate a new TFP series (As), use the new expression for y after substituting the steady state expression for capital per labor unit into the production model, then divide Y (GDP) by L (labor input). Normalize d to equal 1. Calculate the correlation between log A and log As. If it is high, the TFP ranking across countries is similar with both methods; if it is low, the TFP ranking is different.

g. Perform a variance decomposition using log As instead of log A, then find the percentage explained by the two variance terms and the covariance term. The importance of TFP in explaining GDP per labor unit can be concluded based on the results of this variance decomposition, compared to the previous ones.

h. The Solow model helped us understand the importance of TFP by allowing us to analyze the relationship between TFP, capital per labor unit, and savings rate. This allowed us to better understand how TFP affects output and how it is influenced by other factors.

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